The Anharmonic Oscillator

Bachelor thesis
Jordy de Vries

In this article the anharmonic oscillator in quantum mechanics is discussed. First propagators of less complex systems are found by using the path integral method.
With perturbation theory an expression, up to any order, has been found for the vacuum propagator of the anharmonic oscillator. In this article only the first and second order were calculated and by looking at the n-pointfunctions the Feynman diagrams of these orders were found. It appears that the anharmonic term up to first order causes a change in the frequency of the system. Looking at second order perturbation theory gives five diagrams. The first four of these diagrams are simple diagrams and again these diagrams alter the frequency of the system. The change in the frequency looks a lot like a series and it is possible to ’guess’ a general expression up to any order for all simple diagrams. This expression seems to fit qualitatively for higher orders however the prefactors were not checked nor is there a proof formulated. The fifth diagrams is a complex diagram and this diagram has not been solved in a satisfying way.
Finally some recommendations for additional research are given.

Publication date: 
August, 2006